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Collusion

Sotiris Georganas

February 2012

Sotiris Georganas () Collusion February 2012 1 / 31

Outline

1 Cartels2 The incentive for cartel formation

The scope for collusion in the Cournot model3 Game theoretic analysis of self-enforcing collusion

Starting point: Cournot-Nash equilibriumInfinitely repeated gamesGrim trigger strategiesChecking the equilibriumResult: Summary and Theory Discussion

4 Cartels: where, when and how?Profits: When high?Enforcing the agreement: When easier?Expected punishment: When low?Small fluctuations in demand

5 Cartel techniquesOPEC example

6 What to remember from this lectureSotiris Georganas () Collusion February 2012 2 / 31

People of the same trade seldom meet together, even for merrimentand diversion, but the conversation ends in a conspiracy againstthe public, or in some contrivance to raise prices. It is impossibleindeed to prevent such meetings, by any law which either couldbe executed, or would be consistent with liberty and justice.

Adam Smith, The Wealth of Nations, 1776


Cartels

Definition

A cartel is a formal (explicit) agreement among competing firms.

Anticompetitive agreements are observed to exist since ancient times

First named ”cartels” in Germany around 1880 for ”alliances ofenterprises”The name was imported into the Anglophone world during the 1930s

Found to decrease welfare of consumers

Outlawed in the USA by the Sherman Act (1890) although AdamSmith thought it impossible

Public cartels permitted in the United States during the GreatDepression in the 1930s

Continued to exist for some time after World War II in industries suchas coal mining and oil production


Cartels

Cartels also played an extensive role in the German economy duringthe inter-war period.

The word cartel got Anti-German bias in the ’40s, as they were used bythe enemy

Illegal now in most countries (except international cartels)

More frequent when there was no law against them (e.g. Rockefeller’sStandard Oil)Some complicated forms survive, such as the Salary Cap in the NBA

Pattern: some industries are more prone to cartel formation thanothers.


The incentive for cartel formation

It’s the money, stupid!

Market demand functions are downward sloping.

A firm that increases output imposes a negative externality on theother firms in that market by causing the price to drop.

Implication

Uncoordinated firm behavior leads to lower total profits than can beachieved through coordinated behavior.


The scope for collusion in the Cournot model

Recall the Cournot model with two firms producing quantities qi ,i = 1, 2.

Product demand is represented by the inverse demand functionp (q1 + q2).

Firm 1’s profits are given by

π1 (q1, q2) ≡ p (q1 + q2) q1 − C1 (q1) (1)

Insight

When firm 2 increases its output, it reduces firm 1’s profits through theeffect on the market price

∂π1

∂q2= p′ (q1 + q2) q1 < 0 (2)


Recall

At the Cournot equilibrium, each firm maximizes its own profits, given theother firm’s output; hence in equilibrium

∂πi

∂qi= 0, i = 1, 2. (3)

Let Π denote total joint profits: Π ≡ π1 + π2

Insight

At the Cournot equilibrium the effect of an output increase by firm 2 (say)on total profits Π is therefore negative:

∂Π∂q2

∣∣∣∣Cournot eq.

=∂π1

∂q2+

∂π2

∂q2= p′ (q1 + q2) q1 + 0 < 0. (4)

Illustrated in Fig 1.


Total joint profits Π are maximized by the two firms reducing theiroutput levels. Fig 2.


But on the other hand, if the firms do coordinate and reduce theiroutput levels, then each firm will have an incentive to cheat on theagreement by increasing its output: at the coordinated carteloptimum ∂πi/∂qi > 0, for i = 1, 2.Sotiris Georganas () Collusion February 2012 10 / 31

Game theoretic analysis of self-enforcing collusion

Start from the Cournot duopoly model. Two firms in a market.

They compete in quantities.

Firm 1’s quantity: q1.Firm 2’s quantity: q2.

The firms’ profits:

π1 (q1, q2) = p (q1 + q2) q1 − C1 (q1) ,

π2 (q1, q2) = p (q1 + q2) q2 − C2 (q2) .


A Cournot-Nash equilibrium:

(qc1 , qc2 ) is a Cournot-Nash equilibrium if neither firm can increase itsprofits by deviating unilaterally:

π1 (qc1 , qc2 ) ≥ π1 (q1, qc2 ) for every q1,

π2 (qc1 , qc2 ) ≥ π2 (q

c1 , q2) for every q2.

In contrast, the collusive outcome, (qm1 , qm

2 ), maximizes total profits(denoted Π above)

maxq1,q2{π1 (q1, q2) + π2 (q1, q2)} .

and we use πmi to denote the profit obtained by firm i in the collusive

outcome.


The collusive outcome cannot be sustained as a Cournot-Nashequilibrium in a one-shot game.

Suppose Firm 1 expects Firm 2 to produce its share of the optimalcartel output, qm2 . Then Firm 1’s best response would be the solutionto

maxq1

π1 (q1, qm2 ) .

Denote the solution to this problem by qr1.And denote Firm 1’s profit if deviating to qr1 by πr

1 [=π1 (qr1, qm2 )].

We have qr1 > qm1 and importantly

πr1 > πm

1 > πc1 .


Infinitely repeated games

“Infinitely repeated games” are also called “supergames.”

Suppose there is an infinite sequence of time periods: t = 1, 2, 3, . . .In each period, the duopolists simultaneously choose their quantitiesqt1 and qt

2.

They then get the profits π1 (qt1, qt

2) and π2 (qt1, qt

2).

This is repeated in every period, and both players know all previouslychosen quantities.

The list of all chosen quantities prior to period t is called the periodt history of the game [=everything that has happened previously inthe game].


Each firm maximizes the discounted sum of all its future profits. ForFirm 1:

V1 = π1

(q11 , q1

2

)+ δπ1

(q21 , q2

2

)+δ2π1

(q31 , q3

2

)+ δ3π1

(q41 , q4

2

)+ · · ·

=∞

∑t=1

δt−1π1

(qt1, qt

2

),

where δ is a discount factor [recall that δ0 = 1].

Assumption: 0 < δ < 1.

Interpretation:

δ = 11+r , where r is an interest rate.

δ could also reflect the possibility that, with some probability, the gameends after the current period.


Grim trigger strategies

A strategy in a repeated game is a function.

This function specifies, for any possible history of the game, whichquantity a player chooses.

Consider the following “grim trigger strategy” for Firm 1 in period t:

If both firms have played the collusive output (qmi ) in all previousperiods, play the collusive output in this period too.If at least one firm did not play the collusive output (some qi 6= qmi ) inat least one previous period, play the Cournot-Nash output (qci ).


Check if it is an SPNE for the firms to use this strategy:

First we check that no firm has an incentive to deviate unilaterallyalong the equilibrium path. (Requirement for having a Nashequilibrium.)Then we check the same thing off the equilibrium path. (Requirementfor subgame perfection.)

Reminder

The sum of an infinite geometric series:

1 + δ + δ2 + δ3 + · · · = 1

1− δ


Checking the equilibrium

Player 1’s payoff if both players play the grim trigger strategy (fromt = 1 onwards):

V e1 =

∞

∑t=1

δt−1πm1 = πm

1

∞

∑t=1

δt−1

= πm1

(1 + δ + δ2 + δ3 + · · ·

)=

πm1

1− δ.

Payoff if deviating (from the equilibrium path) at t = 1:

V d1 = πr

1 +∞

∑t=2

δt−1πc1 = πr

1 + πc1

∞

∑t=2

δt−1

= πr1 + πc

1

(δ + δ2 + δ3 + · · ·

)= πr

1 + δπc1

(1 + δ + δ2 + · · ·

)= πr

1 +δπc

1

1− δ.


That is, no incentive to deviate if

V e1 ≥ V d

1 ⇔πm1

1− δ≥ πr

1 +δπc

1

1− δ.

i.e. if one period deviation payoff (πr1 − πm

1 ) does not exceed longterm reward from cooperation:

πr1 − πm

1 ≤δ (πm

1 − πc1)

1− δ.

Solving this inequality for δ yields

δ ≥ πr1 − πm

1

πr1 − πc

1

.


Interpretation:

By deviating, you make a short-term gain but get a lower profit in allfuture periods. So if you’re patient enough (sufficiently large δ), thenyou resist the temptation to deviate.


Checking subgame perfection

Any deviation is effectively punished by the competitor.Is carrying out this punishment credible?Imagine that we are in a subgame where at least one firm haspreviously chosen some quantity differing from the collusive output(some qi 6= qmi ).The grim trigger strategy prescribes that then each firm should choosethe Cournot output (qi = qci ).We must verify that this is a Nash equilibrium. Clearly it is!


Result: Summary and Theory Discussion

Conclusion: We can sustain the outcome (qm1 , qm

2 ) (in every period)as an SPNE of the infinitely repeated game if the players caresufficiently much about the future (or, the interest rate r low enough).

However, some potential issues with this equilibrium can be raised.

Equilibrium multiplicity

There are (many) other equilibria: For example, always playing theCournot-Nash quantity is also a SPNE. Multiplicity of equilibria aproblem with this theory — no obvious prediction.The typical approach among IO economists: Assume the firms are ableto coordinate on a collusive equilibrium whenever such exists.

Renegotiation when collusion has broken down - Commitmentproblem

The SPNE where the firms collude is not renegotiation proof: After adefection the firms are supposed to play Cournot-Nash forever. Butboth would be better off if they renegotiated and agreed to forgetabout the deviation and instead start to collude again.


Cartels: where, when and how?

Effective cartels form when:

1 group action can raise price and profits;2 enforcing an agreement is relatively easy;3 the expected punishment is low relative to the gains;4 the fluctuation in demand is low.


CartelsProfits: When high?

Inelastic market demand

Inelastic supply response from non-cartel members and fromproducers of close substitutes.


CartelsEnforcing the agreement: When easier?

When there are few firms in the cartel.

When there is high industry concentration; e.g. a dominant firm cantake the lead.

The product is relatively homogenous: easier to agree price and tomonitor.

There are preexisting inter-firm ties, e.g. trade associations whichcan facilitate information exchange.

Prices are highly visible: e.g. depending on the type of the product.

Players can use violence, e.g. guns (the Sicilian mafia tends to be aninternationally successful cartel)


CartelsExpected punishment: When low?

Expected punishment = probability of detection × punishment.

More strict legislation appears to have had some effect.


CartelsSmall fluctuations in demand

A decrease in the market price may be due to cheating or due to adrop in demand.

Reduces the risk of being detected when cheating.


Cartel techniques

Some techniques are well-known:

Geographic- or quota sharingMost-favored-customer clause. Contractual commitment by a sellerthat all customers will pay the lowest price charged any customer.

Makes it more costly for a firm to deviate from a collusive agreement(since it has to charge the lower price to all customers).Gives the customers a stronger incentive to watch out for (secret) pricecuts, which then also the competitors can find out about.

Meeting competition clauses (Never knowingly undersold): A way ofobtaining information.


Cartel techniquesOPEC example

OPEC is a quite powerful cartel. How does it work?

Why is the cartel sometimes unstable?


Issues we did (and will) not discuss

Tacit collusion (read Tirole, Chapter 6)

General antitrust issues

What is the relevant market?


What to remember from this lecture

What is collusion and why is there scope for collusion in the standard(Cournot-Nash) duopoly model.

The logic behind how repeated interaction can make collusionself-enforcing when the firms are patient.

Differences between cartel models and real life cartels

Circumstances and practices that facilitate the sustainability of cartels.


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